Let \lambda be a partition of n. We consider the Kazhdan-Lusztig basis of the corresponding Specht module, which is indexed by standard Young tableau of shape \lambda. One of the amazing features of this basis is that it can be used to relate representation theoretic properties of Specht modules to combinatorial properties of tableau. For example, in the 90s Berestein-Zelevinsky and Stembridge showed that the long element of the symmetric group acts on the Kazhdan-Lusztig basis by the Schutzenberger involution on tableau. Similarly, in 2010 Rhoades showed that the long cycle (1,2,...,n) acts by the jeu de taquin promotion operator when \lambda is rectangular. In this talk we will explain how to use braid groups acting on triangulated categories to generalize Rhoades' result in three directions: we lift the condition on the shape of the partition, we greatly enlarge the class of permutations for which the result holds, and we prove analogs in other Lie types. This is based on joint work with Martin Gossow.